Gravity Drilling to the Center of the Earth

If you were to drill a hole through the center of the earth the result might seem a little counter intuitive.

Gravity decreases with distance, away from the earth, ie in outer space, and it also decreases as you travel toward the center of the earth.

The reason is that the force of gravity on an object will only come from the inner layers (relative to position of the object).

The volume of the sphere with radius of the distance from the center will affect you, not any outside layers, since the outside layers will cancel forces.

The part of the outside layer near to you will be closer with less mass and the part farther away will have more mass (proportional inversely to square of distance).

Newton Derives Universal Gravitation Constant and Mass of Earth

Skeptics did not believe Newton’s theories of gravitation to be true, but his figures proved valid.

The radius of the Earth has been known since ancient times and the distance to the moon has also been calculated.

Therefore Newton could use these numbers in a clever way to calculate both the universal gravitation constant and the mass of the earth.

When he plugged in numbers it showed that they resulted in the period of the moon being 1/13 of a year, which is accurate.

In a similar way you can often times manipulate multiple variables using known data (like using the calculus technique of summing shells to obtain a volume) to obtain solutions.

Natural Frequency Introduction

All objects that oscillate have something called a natural frequency.

At this frequency they will experience resonance and start oscillating with increasing magnitude towards infinity, or in the case of something like a bridge until it breaks.

In another instance, a snare drum can be sitting motionless, but if a saxophone player near it plays a certain tone that happens to match its natural frequency (or a multiple of it) then the snare drum will start vibrating.

Physics Mathematics Notation Euler Sine Cosine

When writing mathematical expressions, numbers and formulas that look quite different can mean the same thing.

For instance, power series (an infinite series of terms added together) can be used to represent things like the natural log (ln).

That is actually how devices such as calculators compute numbers.

Likewise e with imaginary exponents is used as part of an expressions by Euler for sine and cosine.

Pendulum Motion and Simple Harmonic Approximation

Technically the movement of a pendulum doesn’t quite match simple harmonic motion.

However, if the angle theta is small enough,  θ~sin(θ)

In fact for angles less than 15 degrees the error will be less than one percent.

Thus a simple pendulum essentially displays SHM (simple harmonic motion).

Such approximations are very useful in certain applications, especially in engineering, where such a small amount of error can be tolerable.

Fear and Physics

From mozzercork on flickr

(from mozzercork on flickr)

If you have a tough physics problem, you might hesitate to even start it.

That won’t help your cause though…..

Use lots of paper and just write down what comes to your mind.

It might be completely wrong, but it doesn’t matter, you’ll find out pretty quickly by doing simple checks (dimensional analysis, feasibility, etc)

Thomas Edison tried 10,000 materials for filaments before getting the electric light to work.

And complex physics problems can get somewhat ridiculous.

If you start a problem earlier, it will be in the back of your head.

Professor Barber even told us that he has gone to sleep and woken up with the answer.

Tension Free Body Diagram

If a vine is hung over a tree branch in the forest and two monkeys are hanging on both ends then the tension in the vine will be equal throughout the whole rope, since if it wasn’t the vine would snap or go slack.

The tension force for one monkey’s side will be equal and opposite to the tension force on the other monkey’s side.

The vine itself will also require a different amount of force to slide across the top of the branch depending on how the vine and the tree branch interact.

The force required for movement is unique depending on the materials (which defines a coefficient of friction).

Objects will have different coefficients of friction, denoted as mu, based on how they act together.

In analyzing situations it is good to draw a picture or “free body diagram” and decide what the axes are, including which directions should be understood as positive and negative.

The free body diagram should  only be applied to simple situations (generally one or a few objects involved).

When the situation is complicated, like the effect of huge numbers of electrons hitting an object, other theories come into play.

Even with fifty forces that are three dimensional, you would have to add up 150 components (x, y, z for each) and that wouldn’t be fun.

Simple Harmonic Motion SHO

Many things exhibit something that resembles simple harmonic motion (abbreviated SHO).

Waves rise and fall with a regular pattern and a slinky oscillates.

To analyze simple harmonic motion, we turn to the sine and cosine functions which are also periodic.

With a bit of tweaking, according to different amplitudes and starting points, a harmonic motion equation can describe such periodic functions.

However, such methods only work for small displacements.

When the displacements get large enough, nonlinear effects come into play.

Phil Kesten Physics 32 Class at Santa Clara University

Hi Neal,
Okay, the pressure’s on:  I told my students about your site!  And yes, I’m teaching 32 this quarter, so anything you add on SHM, fluids, gravitation, waves, light, sound…  they’ll be looking!

prk

In light of this news, I will start to talk about Physics 32 stuff.

As far as my connection to Dr. Kesten, he was my advisor at Santa Clara and I took physics 31, 32, 33, and 34 from him.

Right now I am doing an MS in applied physics.

If I am unclear, let me know and leave comments!

I can even make videos, but only if I feel it is worth my time for now.

Changing Coordinate Systems to Make Problems Easier

One day a physicist comes across a perplexed student in the library.

It seems that the student had only learned to place his axes with the positive y axis going up and the positive x axis going to the right.

Unfortunately this system wasn’t working so well with a situation involving pulleys and curved directions.

The physicist decides to help the engineer by telling him that the x-y coordinate system can be changed to better suit the problem at hand, much like polar coordinates can be more convenient than cartesian coordinates.

If an object travels in a curved path he reasons that the x-y coordinate system may also be curved in parts for simplicity.

Gravity may be thought of as positive or negative and up may be considered down.

The physicist also says that simply using common sense to check an answer will often times be effective, for instance if a block is sitting on a wall with no forces it will probably just stay there.